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The concentration-based ionic strength is $I_c=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$, where $c_i$ is each dissolved ion's concentration and $z_i$ its signed electric charge number. It has the same units as $c_i$; some formulas instead use the dimensionless ratio $I_c/c^{\circ }$. All dissolved ions contribute, including those that balance the electric charge of a buffer solution's conjugate base. Ionic strength helps describe the interactions represented by activity coefficients. See Hippler and Metcalfe, activity coefficients and dimensionless ionic strength.

The ionosphere is a region of the upper atmosphere of Earth with substantial partial ionization. Its mobile electrons and ions constitute a plasma mixed with neutral gas. For radio communication, the crucial question is whether a chosen radio wave returns toward the ground, is absorbed, or passes through. A skywave can connect stations beyond their direct view across the curved surface of Earth, while wave interference between a ground wave and a skywave can make reception fade even when the radio transmitter remains steady.

The ionosphere is useful both as a route for radio communication and as a remote laboratory: varying the probing frequency changes how deeply the radio wave penetrates the plasma. The Appleton-Barnett radio interference experiment turned an unwanted fluctuation in reception into a measurement of the overhead route. Cold-plasma radio dispersion explains why this is a tunable medium, while ionospheric virtual height explains why a measured delay is not automatically an altitude.

An ionotropic receptor is a receptor in which binding of a ligand directly gates an ion channel within the same protein complex. The binding site detects the chemical signal; the pore supplies a path across the cell membrane. At a chemical synapse, a neurotransmitter therefore changes synaptic conductance without an obligatory intervening G protein or second messenger. ^[ref]

A familiar example is the nicotinic acetylcholine receptor of skeletal muscle: acetylcholine binding opens a pore passing cations, including sodium and potassium. The muscarinic acetylcholine receptor instead uses intracellular signal transduction; the same neurotransmitter can engage different mechanisms. Direct gating does not mean that an ionotropic receptor pumps ions: electrochemical gradients drive passage through the open ion channel. Nor does “ionotropic” specify excitation. The reversal potential, initial membrane potential and other conductances determine whether the effect favors an action potential, causes hyperpolarization, or supplies shunting inhibition. ^[ref]

A concrete electrical approximation separates opening from drive. Let $N$ identical ion channels each have open-channel electrical conductance $\gamma$ and opening probability $p(t)$. At fixed membrane potential $V$, with common reversal potential $E_{\mathrm{rev}}$, the expected value of inward-positive electric current isFor stipulated values $N=100$, $\gamma =20\mathrm{pS}$, $p=0.2$, $V=-60\mathrm{mV}$ and $E_{\mathrm{rev}}=0$, the mean open count is $20$, the population electrical conductance is $0.4\mathrm{nS}$ and $I_{\mathrm{in}}=24\mathrm{pA}$. One open ion channel carries $1.2\mathrm{pA}$. These are illustrative values, not measurements. At $V=E_{\mathrm{rev}}$, the net electric current vanishes despite open ion channels; neither binding nor opening alone determines the readout. The approximation omits changes in ion concentrations and voltage-dependent open-channel electrical conductance.

A motivating experiment made the elementary electrical events visible. As Bert Sakmann recounts, chronically denervated frog muscle fibers had acetylcholine-sensitive receptors spread beyond the original nerve contact. Mild enzyme treatment exposed the cell membrane; a polished glass pipette sealed over a small patch to reduce background noise. Two intracellular electrodes held the local membrane potential through voltage clamp, while the patch pipette recorded tiny electric currents. Applying suberyldicholine, a ligand that activates the acetylcholine receptor and prolongs openings, revealed abrupt, approximately rectangular current events. Their amplitudes agreed reasonably with estimates from whole-membrane current fluctuations. This established discrete opening and closing events; it did not make every ligand response instantaneous. Background noise and finite recording bandwidth concealed sufficiently small or brief events.^[ref]

Recording an intact membrane patch alone does not exclude intracellular relays. Tank, Huganir, Greengard and Webb supplied a complementary isolation test in 1983: they purified nicotinic acetylcholine receptors from an electric ray's electric organ, incorporated them into artificial lipid membranes, and fused small vesicles into larger vesicles by freezing and thawing. Sealed glass-pipette recordings from isolated membrane patches resolved functional ion channels with properties resembling those in muscle. Removing the intact cytoplasm and its signaling machinery supports the sufficiency of the receptor complex for channel function; an artificial lipid environment still cannot reproduce every native regulatory interaction.^[ref]

Direct gating supports millisecond communication at many chemical synapses. Response duration also depends on neurotransmitter clearance, binding and channel-state transitions. Continued ligand exposure can prolong activity. Classify an ionotropic receptor by its direct connection between recognition and pore gating, rather than a universal duration cutoff. ^[ref]

An ion pair is an associated cation and anion. An isolated ion pair lacks the extended neighbourhood of an ionic crystal. In a chemical solution, ion pairs can coexist with more widely separated ions; their abundance depends on conditions. A neutral gas-phase sodium chloride molecule can be treated as a discrete entity, with its own quantum mechanics and dissociation products. That use of molecule does not describe the constituents of ordinary solid sodium chloride.^[ref][ref]

Spatial variation of the ion temperature. The inverse logarithmic scale in the title is positive for an outward-decreasing profile. A nonzero value is a possible free energy source; it does not by itself establish ion-temperature-gradient instability without the appropriate dynamics and threshold.

A drift-scale plasma instability fed by an ion temperature gradient. In common electrostatic limits, Boltzmann electron response, ion temperature evolution, magnetic drifts and polarization drift determine the dynamics. Unfavorable magnetic field-line curvature can drive a toroidal branch; straight-field branches can involve parallel compression. The threshold depends on the equilibrium gradients and geometry.

An ion trap confines an ion using electromagnetic fields, allowing repeated interrogation of the same particle. The name also covers related traps for an electron. A Paul trap uses an oscillating electric quadrupole field; a Penning trap combines a static electric quadrupole field with a static magnetic field. Confinement creates time to measure an energy level, frequency or magnetic dipole moment; it does not itself cool the particle or identify its internal quantum state.

The obstacle is Earnshaw's theorem. In a charge-free neighbourhood the static electric potential $\Phi$ obeys Laplace's equation, ${\nabla }^2\Phi =0$. For a particle of fixed nonzero electric charge $Q$, the potential energy $Q\Phi$ is also a harmonic function. The maximum principle for harmonic functions excludes a strict interior minimum. For a quadratic example, the Hessian matrix has zero trace; it cannot have three positive eigenvalues. A static purely electric field therefore cannot supply three-dimensional stable confinement of a free point electric charge. Time variation in a Paul trap, and the velocity-dependent Lorentz force in a Penning trap, change the assumptions of this argument.

An IP3-sensitive calcium store contains releasable calcium behind a membrane bearing functional IP3 receptors. The endoplasmic reticulum is an important example. Pumping calcium into it or exporting calcium across the cell membrane lowers free cytosolic calcium; intracellular redistribution and loss from the whole biological cell are different balances.^[ref]

Finite availability matters: a release pathway cannot deliver more calcium than its accessible store contains unless replenishment occurs. A falling cytosolic calcium response can reflect reduced release, increased removal, or both. Store content, release rate and free cytosolic concentration therefore answer different questions. Binding to proteins can also change the free fraction without changing the total number of calcium ions.

Isotopes of a chemical element have the same atomic number $Z$, hence the same number of protons, but different neutron numbers $N$ and therefore different mass numbers $A=Z+N$. For example, every neon atomic nucleus contains ten protons: neon-20, neon-21, and neon-22 contain ten, eleven, and twelve neutrons, respectively. They occupy one chemical element box in the periodic table, rather than three boxes.

Three labels answer different questions. The chemical element fixes $Z$; the isotope specifies which neutron number accompanies that $Z$; the ion label specifies the net electric charge. Neutral ${}_{10}^{22}\mathrm{Ne}$ and the ion ${}_{10}^{22}{\mathrm{Ne}}^{+}$ both contain ten protons and twelve neutrons; they contain ten and nine electrons, respectively. Ordinary ionization changes the electron count, leaving the isotope unchanged. Changing the neutron number can change nuclear binding energy and the possibility of radioactive decay. Isotopic abundances can help distinguish material sources, while radioactive decay can additionally supply chronological information.

A mass spectrum makes the distinction experimentally visible: a single chemical element can give separate ion peaks at different mass-to-charge ratios. Its bulk atomic weight is a weighted mean governed by isotopic abundances, rather than the location of one compulsory peak. The historical mass spectrograph experiment and the synthetic neon mass spectrum below illustrate these complementary observations.

Replacement of a lattice ion by another of the same formal charge. charge neutrality alone does not ensure unlimited solubility; size and coordination preferences also matter.

A lattice energy needs a reaction, a sign convention, and a stated amount. Using the lattice-dissociation enthalpy convention for sodium chloride,The reverse lattice-formation enthalpy is negative with the same magnitude when initial and final states match. Chemistry 2e gives $769\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for this separation, per mole of sodium chloride formula units. This is about $7.97$ electronvolts per formula unit, not per individual ion. It concerns separated gaseous ions, rather than neutral chemical atoms, molten salt, or hydrated ions in water.^[ref]

Also distinguish an internal energy change from an enthalpy change: $H=U+pV$. A static point-charge calculation omits motion and thermal contributions. Reversing a reaction reverses its energy change; replacing internal energy with enthalpy additionally changes the defined quantity. The Born-Haber cycle infers a lattice-formation enthalpy by combining other enthalpies, ionization energies, and electron affinities. These imagined stages are a thermodynamic accounting path, not a claim about the actual sequence of a chemical reaction.^[ref]

A second indirect route makes the experimental distinction especially clear: a calorimeter can measure dissolution of a weighed salt into a known amount of water, but its final ions are hydrated, not separated gaseous ions. At compatible standard states and the same temperature,Here the combined hydration of ions term takes one mole each of gaseous ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ to the specified aqueous reference state. Its sum suffices: dissolution calorimetry alone cannot independently assign single-ion hydration of ions enthalpies, or determine $L_{\mathrm{diss}}$ without another input. A finite-concentration measurement also requires dilution corrections before it can supply $\Delta H_{\mathrm{sol}}^{\circ }$.

In a simple constant-pressure solution calorimeter, an insulated vessel, stirrer and thermometer follow the temperature change after mixing. A calibrated effective heat capacity $C_{\mathrm{eff}}$ converts the corrected temperature change to heat. With no applied heating, negligible external heat transfer, and only pressure-volume work, the approximate molar result isHere $n$ is the amount of substance dissolved and $C_{\mathrm{eff}}$ includes the chemical solution and the vessel. Cooling means positive dissolution enthalpy, because the dissolving sample takes heat from its surroundings.^[ref]

An actual reference measurement used potassium chloride, not sodium chloride: a platinum-lined calorimeter with an adiabatic shield and a quartz-oscillator thermometer. Because dissolution absorbed heat, measured electrical heating prevented loss of shield control; the readouts were electrical input and the vessel/shield temperature record. Stirring was checked visually in a dummy vessel. The thermometer was compared with a calibrated resistance thermometer, and electrical calibration bracketed the reaction. The reported result was $+(17.584\pm 0.017)\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at $298.15\mathrm{K}$, for one mole of potassium chloride in $500$ moles of water.^[ref]

Checks on residual water, particle size and final concentration, and agreement with a second calorimeter, constrained systematic errors. Imperfect shield tracking required heat transfer corrections. This experiment measures an enthalpy of solution, not a direct lattice energy; the hydration and dilution bookkeeping still has to be supplied. The full article could not be opened here; the inspected indexed text supports the stated apparatus and checks, without reconstructing its complete uncertainty budget.^[ref]

Ernest Orlando Lawrence and Milton Stanley Livingston reported repeated ion acceleration in 1932. Their evacuated brass chamber used one driven hollow cyclotron dee; the grounded box and slotted divider supplied the other electrode. Filament electrons ionized hydrogen near the gap. A perpendicular magnetic field bent the ions; a high-frequency oscillator drove the gap. Deflecting plates and slits selected particles for a Faraday cup, read by an electrometer.^[ref]

Scanning the magnetic field produced narrow resonance peaks. The optimum deflecting voltage tracked the kinetic energy predicted from field and orbit radius, supporting the reported $1.22\mathrm{MeV}$ protons. Source width and slit geometry limited this check; it was not a direct energy measurement of every particle.^[ref]

This combination distinguishes producing a circulating electric current from establishing its kinetic energy. The electrometer measures collected electric charge per unit time; it cannot by itself tell whether the collected particles are fast. The separate deflection selection supplies a speed-dependent test. Agreement of two consequences of the model, the cyclotron resonance condition and the cyclotron energy and orbit radius relation, makes the inference stronger than either a current signal or a predicted orbit alone.

Milton Stanley Livingston built the early cyclotrons with Ernest Orlando Lawrence; the 1939 Nobel Prize in Physics went to Ernest Orlando Lawrence. His lecture, The evolution of the cyclotron, was delivered on 11 December 1951.^[ref][ref]

Before calculating electronic energy, ask a simpler question: where can the available valence electrons go? A Lewis structure makes that budget visible, and nitrate shows why a correct budget can still require several resonance structures.

A Lewis structure records which chemical atoms are connected in a molecule or ion and how its valence electrons are allocated. One line represents one bond pair, two lines two bond pairs, and three lines three bond pairs; two dots beside a chemical atom represent one lone pair. The marks are a counting language, rather than measured electron positions. See the IUPAC Gold Book, Lewis formula.

For water, ${\mathrm{H}}_2\mathrm{O}$, the budget is $6+2\times 1=8$ valence electrons. Two O–H lines use four electrons; the other four form two lone pairs on oxygen. For ammonia, ${\mathrm{NH}}_3$, $5+3\times 1=8$ valence electrons give three N–H lines and one lone pair on nitrogen. Both molecules contain four electron pairs, but their allocations differ. Each shared bond pair is counted once in the total budget, even though both endpoints count it when checking the octet rule.

A practical construction method is:

For carbon dioxide, $4+2\times 6=16$ valence electrons. Starting with two single C–O connections leaves twelve electrons, enough for three lone pairs on each oxygen, but only four electrons around carbon. Converting one lone pair from each oxygen into a second bond pair gives $\mathrm{O}=\mathrm{C}=\mathrm{O}$. Eight electrons are in the four bond pairs, and eight are in four lone pairs, two on each oxygen. Carbon has no lone pair; all three chemical atoms satisfy the octet rule, and all three formal charges vanish.

For the nitrate ion, ${\mathrm{NO}}_3^{-}$, the valence electron budget isNitrogen contributes five, the three oxygen chemical atoms contribute eighteen, and the negative electric charge adds one electron. Start with nitrogen connected to each oxygen by one bond pair: those three lines use six electrons. Put the remaining eighteen electrons into three lone pairs on each oxygen. Each oxygen now satisfies the octet rule, but nitrogen counts only six electrons. Reassign one oxygen lone pair as a second bond pair on its N–O connection. This changes the allocation without adding an electron; it applies the construction procedure in Flowers et al., Chemistry 2e, 7.3: drawing Lewis structures.

The resulting nitrate Lewis structure has one N=O connection and two N–O connections. Its four bond pairs use eight electrons, and its eight lone pairs use sixteen: $8+16=24$. The doubly connected oxygen has two lone pairs; each singly connected oxygen has three; nitrogen has none. All four chemical atoms satisfy the octet rule. Any of the three oxygen chemical atoms can supply the additional bond pair, giving the three equivalent resonance structures below.

Why not draw three N=O connections to make the ion look symmetric? With two lone pairs on each oxygen, that drawing still spends $12+12=24$ electrons and has the right total formal charge, but places twelve electrons around nitrogen. It fails the octet rule for this second-period chemical element. Three equivalent resonance structures express the symmetry without that invalid local count.

The economy is beautiful: a handful of dots and lines rules out many impossible counts and identifies lone pairs that can participate in acid-base chemistry. A Lewis structure helps choose rational candidate connections before a detailed quantum chemistry calculation. It does not specify a complete many-electron wavefunction, actual electron density, or molecular geometry. Adding the separate VSEPR model, described in Chemistry 2e, 7.6: molecular structure and polarity, predicts bent water and pyramidal ammonia; the layouts above alone make no such prediction. Molecular orbital theory supplies a richer electronic description, including the unpaired electrons of molecular oxygen that an all-paired Lewis structure misses.

In 1916 Gilbert N. Lewis proposed two-dot notation for the shared electrons of a chemical bond and fuller dot formulae for the outer electrons, explicitly relating a shared pair to the existing single-line notation. See Lewis, The Atom and the Molecule, JACS 38, 762–785 (1916), pp. 777–778, available as an ^[ref]; the relevant passage is also readable in a ^[ref]. The modern quantum mechanics interpretation should be distinguished from his historical model of fixed outer electrons.

A biochemical ligand is an ion or molecule that binds a protein. For a receptor, a ligand may activate signaling, prevent activation, or have another effect. This usage concerns protein binding; it is distinct from the coordination-chemistry use of the word. See the protein-binding reference under receptor.

A motivating experiment used isolated membranes from A-431 biological cells, which supplied abundant EGFR. Separate membrane aliquots were incubated at $0{}^{\circ }\mathrm{C}$ with ATP carrying radioactive phosphorus in its terminal phosphate group, plus magnesium or manganese ions. Radioactivity in material recovered by chemical acid-induced precipitation supplied the readout. Adding EGF beforehand enhanced incorporation relative to the unstimulated preparation; this placed a ligand-dependent biochemical response in a cell-free membrane system.^[ref]

The follow-up comparisons separated increased addition from suppressed removal: phosphate removal was rapid but unaffected by EGF, and anti-EGF antibody reversed activation by removing the ligand. These observations support reversible regulation of membrane-associated protein kinase activity.^[ref] Total incorporation alone cannot identify which protein or amino acid residue received the label, prove intrinsic receptor catalysis, or predict an intact biological cell's response.

The same account records an instructive identification error: receptor phosphorylation was initially assigned to the wrong amino acid residue because released modified amino acids migrated together in the electrophoretic electric field. Reanalysis identified phosphotyrosine residues. Retained radioactivity is evidence about the assay's separation and detection, not unique chemical identity.^[ref]

A ligand in coordination chemistry is an atom, ion or molecule attached to a central atom through one or more donor sites. This use is distinct from a protein-binding ligand. A neutral CO ligand and an anionic chloride ligand can both donate two electrons in ionic electron counting.

For a modern chosen example, the National Institute of Standards and Technology lists neutral magnesium atomic masses $23.985041697\mathrm{u}$, $24.985836976\mathrm{u}$, and $25.982592968\mathrm{u}$, with reference isotopic abundances $0.7899$, $0.1000$, and $0.1101$. They describe the isotopes with mass numbers 24, 25, and 26, respectively.^[ref] These modern inputs are not a reconstruction of historical observations.

Use the dimensionless mass-to-charge ratio coordinate $x=(m/\mathrm{u})/z$. For the illustration only, approximate ion masses by the neutral atomic masses. The singly charged positions are then near $23.98504$, $24.98584$, and $25.98259$; the doubly charged companions are near $11.99252$, $12.49292$, and $12.99130$. The neutral-to-ion correction already given above is essential at higher accuracy. A peak near $x=12$ alone need not identify a chemical element: another ion with a different electric charge can occupy the same mass-to-charge ratio interval.

For the narrow singly charged peak at $x\approx 24.98584$, the chosen full width at half maximum $\Delta x=0.12$ gives mass resolving power $R\approx 208.2$; widening to $\Delta x=1.20$ gives $R\approx 20.82$. Adjacent isotope spacings here are approximately one unit in $x$. Peaks are visibly separated in the first chosen profile and overlap strongly in the second. A small uncertainty in a fitted peak centre would not by itself make the overlapping components resolved.

Even integrated peak areas need a response model. For isotope fraction $f_i$, let ${\eta }_{i,z}$ collect the probability of ionization into electric charge state $z$, transmission through the analyzer, and the detection response. In a linear instrument with overall exposure factor $C$, the component area isEqual ${\eta }_{i,z}$ at a fixed $z$ makes normalized areas equal the fractions; otherwise it does not. In the chosen example, doubling only the mass-25 response changes the normalized areas to $0.71809$, $0.18182$, and $0.10009$, although the neutral sample is unchanged. Photographic darkening introduces another possible nonlinearity: neither line darkness nor peak height is a universal measure of neutral isotopic abundance. Calibration of position and calibration of intensity answer different questions. Matched standards, a known electric charge assignment, and a response correction make isotope analysis useful; a visually impressive set of peaks alone does not supply those controls.

An ionic compound containing divalent magnesium and sulfate ions. Its nominal formula concentration $c$ contributes ionic strength $4c$ when fully dissociated into these two ion species.

Strong longitudinal magnetic guidance can average rapid electron gyro-motion during relevant ion-electron interactions. The effective ion equilibrium scale can then be much lower than the transverse electron gyro-temperature, subject to gyroradius, collision-scale and beam-quality limits.